===== Complex exponents with positive real bases =====
==== Function ====
| @#55CCEE: context     | @#55CCEE: $ b\in\mathbb R_+^* $ |
| @#FFBB00: definiendum | @#FFBB00: $ z\mapsto b^z :\mathbb C\to\mathbb C $ |
| @#FFBB00: definiendum | @#FFBB00: $ z\mapsto b^z := \mathrm{exp}(z\cdot \mathrm{ln}(b)) $ |

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=== Discussion ===
The identity 

$b^{x_1+x_2}=b^{x_1}\cdot a^{x_2}$,

says that exponentiation is a (the) homomorphism between $+$ and $\cdot$.

The combinatorial manifestation, e.g. formulated in for $B,X_1,X_2,\dots\in\bf{Set}$, is

$B^{\coprod_{j\in J}X_j}\cong\prod_{j\in J} B^{X_j}$

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=== Context ===
[[Natural logarithm of real numbers]]